1. Engaging Students in Learning Multiplication 6th Grade
Pam Hutchison
February 6, 2017

2. Agenda Statistics Geometry Measures of Center Measures of Variability
Comparing Data
Geometry
Area
Volume

3. Statistics

4. Measures of Center
Mode
Median
Mean

5. Measures of Variability
Range
Interquartile Range
Mean Absolute Deviation (MAD)

6. Concrete What’s in a Hand?
Students reach in and take a handful of multilink cubes
Link them together into a tower
Line up from smallest to tallest

7. List the data in order from smallest to largest
12, 13, 13, 14, 14, 17, 17, 17, 18

8. Minimum Value: 12 12

9. 18
Maximum Value: 18

10. 18
Range
= 6 12

11. Frequency Table Number Number of of Cubes Students 12 1 13 2 14 2 17 3
12 1
13 2
14 2
17 3
18 1

12. Dot Plot | | | | | | |
    

13. 17
Mode: 17

14. Relist the data (in order)
12, 13, 13, 14, 14, 17, 17, 17, 18

15. Median
= 14

16. X X X X X X X X Dot Plot | | | | | | |     
How could you find the median?
| | | | | | | X
 X X X   X X X X  

17. Relist the data (in order)
12, 13, 13, 14, 14, 17, 17, 17, 18
List the data from the lower half
12, 13, 13, 14
Find the median 13
This is called the 1st Quartile

18. Relist the data (in order)
12, 13, 13, 14, 14, 17, 17, 17, 18
Now list the data from the upper half
17, 17, 17, 18
Find the median 17
This is called the 3rd Quartile

19. This is called the Interquartile Range (IQR)
12, 13, 13, 14, 14, 17, 17, 17, 18
Median: 14
1st Quartile: 13
3rd Quartile: 17
What’s the difference between the 1st quartile and the 3rd quartile?
17 – 13 = 4
This is called the Interquartile Range (IQR)

20. Box and Whisker Plot | | | | | | | 12 13 14 15 16 17 18 Minimum Value: 
| | | | | | |
Minimum Value: 12
Maximum Value: 18

21. Box and Whisker Plot | | | | | | | 12 13 14 15 16 17 18 Median: 14 
| | | | | | |
Median: 14
1st Quartile (Lower Quartile): 13
3rd Quartile (Upper Quartile): 17

22. Box and Whisker Plot | | | | | | | 12 13 14 15 16 17 18 
| | | | | | |
Draw a box using the upper and lower quartiles
Draw whiskers from the lower quartile to the minimum and from the upper quartile to the maximum

23. Box and Whisker Plot | | | | | | | 12 13 14 15 16 17 18 17 – 13 = 4 
| | | | | | |
17 – 13 = 4
What is the interquartile range?

24. Mean
= 15

25. Area

26. G.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

27. Unpacking the Standard
Students continue to understand that area is the number of squares needed to cover a plane figure. Students need to know the formulas for rectangles, triangles, and other quadrilaterals. “Knowing the formula” does not mean memorization of the formula. To “know” means to have an understanding of why the formula works and how the formula relates to the measure (area) and the figure. This understanding should be for all students.

28. G.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
Why a rectangle?

29. Parallelograms
Can you find a way to turn your parallelograms into rectangles?
Scissors, tape, grid paper
How can you use what you just discovered to develop a formula for finding the area of a parallelogram (without having to cut it up)?

30. Parallelograms

31. Parallelograms

32. Parallelograms

33. Triangles
Can you find a way to turn your triangle (or triangles) into a rectangle or a parallelogram?
Scissors, tape, grid paper
How can you use what you just discovered to develop a formula for finding the area of a triangle (without having to cut it up)?

34. Triangles – Version 1

35. Triangles – Version 2a

36. Triangles – Version 2a

37. Triangles – Version 2b

38. Triangles – Version 2b

39. Triangles – Version 3

40. Triangles – Version 3

41. Trapezoids
Can you find a way to turn your trapezoid (or trapezoids) into rectangles, parallelograms, or triangles?
Scissors, tape, grid paper
How can you use what you just discovered to develop a formula for finding the area of a trapezoid (without having to cut it up)?

42. Trapezoids – Version 1

43. Trapezoids – Version 2

44. Trapezoids – Version 2

45. Trapezoids – Version 3

46. Volume

47. G.2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

48. Volume Example:
V = base area x height
V = x
V = 60 cubic cm 15 4

49. V = 60 cubic cm Volume Example: V = length x width x height V = x x 53 4

50. V = 88.2 cubic cm Volume Example: V = base area x height V = x 17.64 5

51. V = 88.2 cubic cm Volume Example: V = length x width x height
V = x x 5
V = 88.2 cubic cm

52. Box of Clay
A box 2 centimeters high, 3 centimeters wide, and 5 centimeters long can hold 40 grams of clay. A second box has twice the height, three times the width, and the same length as the first box. How many grams of clay can it hold?